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1: 32.1 Special Notation

§32.1 Special Notation

►(For other notation see Notation for the Special Functions.) …

2: 6.1 Special Notation

§6.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►Unless otherwise noted, primes indicate derivatives with respect to the argument. …

3: 1.1 Special Notation

§1.1 Special Notation

►(For other notation see Notation for the Special Functions.) …

4: 24.1 Special Notation

§24.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►

Bernoulli Numbers and Polynomials

… ►Among various older notations, the most common one is … ►

Euler Numbers and Polynomials

…

5: 4.1 Special Notation

§4.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. …

6: 5.1 Special Notation

§5.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►The notation $\mathrm{\Gamma }\left(z\right)$ is due to Legendre. Alternative notations for this function are: $\mathrm{\Pi }(z-1)$ (Gauss) and $(z-1)!$. Alternative notations for the psi function are: $\mathrm{\Psi }(z-1)$ (Gauss) Jahnke and Emde (1945); $\mathrm{\Psi }(z)$ Davis (1933); $𝖥(z-1)$ Pairman (1919). …

7: 12.1 Special Notation

§12.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►These notations are due to Miller (1952, 1955). …The notations are related by $U(a,z)=D_{-a-\frac{1}{2}}\left(z\right)$. Whittaker’s notation $D_{\nu }\left(z\right)$ is useful when $\nu$ is a nonnegative integer (Hermite polynomial case).

8: 9.1 Special Notation

§9.1 Special Notation

►(For other notation see Notation for the Special Functions.) ► ►►

$k$	nonnegative integer, except in §9.9(iii).
…

… ►Other notations that have been used are as follows: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi }\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi }\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-1/3}\pi \mathrm{Ai}\left(-3^{-1/3}x\right)$ (Szegő (1967, §1.81)); $e_0(x)=\pi \mathrm{Hi}(-x)$, ${\tilde{e}}_0(x)=-\pi \mathrm{Gi}(-x)$ (Tumarkin (1959)).

9: 8.1 Special Notation

§8.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►Unless otherwise indicated, primes denote derivatives with respect to the argument. … ►Alternative notations include: Prym’s functions $P_z(a)=\gamma (a,z)$, $Q_z(a)=\mathrm{\Gamma }(a,z)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma (a+1,z)$, $[a,z]!=\mathrm{\Gamma }(a+1,z)$, Dingle (1973); $B(a,b,x)={\mathrm{B}}_x(a,b)$, $I(a,b,x)=I_x(a,b)$, Magnus et al. (1966); $\mathrm{Si}(a,x)\to \mathrm{Si}(1-a,x)$, $\mathrm{Ci}(a,x)\to \mathrm{Ci}(1-a,x)$, Luke (1975).

10: 7.1 Special Notation

§7.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►Unless otherwise noted, primes indicate derivatives with respect to the argument. … ►Alternative notations are $Q(z)=\frac{1}{2}\mathrm{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\mathrm{\Phi }(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$, $\mathrm{Erf}z=\frac{1}{2}\sqrt{\pi }\mathrm{erf}z$, $\mathrm{Erfi}z={\mathrm{e}}^{z^2}F\left(z\right)$, $C_1(z)=C\left(\sqrt{2/\pi }z\right)$, $S_1(z)=S\left(\sqrt{2/\pi }z\right)$, $C_2(z)=C\left(\sqrt{2z/\pi }\right)$, $S_2(z)=S\left(\sqrt{2z/\pi }\right)$. ►The notations $P(z)$, $Q(z)$, and $\mathrm{\Phi }(z)$ are used in mathematical statistics, where these functions are called the normal or Gaussian probability functions. …